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Higher rank quantum-classical correspondence

Joachim Hilgert, Tobias Weich and Lasse L. Wolf

Vol. 16 (2023), No. 10, 2241–2265
Abstract

For a compact Riemannian locally symmetric space ΓGK of arbitrary rank we determine the location of certain Ruelle–Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle–Taylor resonances and establish a spectral gap which is uniform in Γ if GK is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e., a one-to-one correspondence between horocyclically invariant Ruelle–Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on GK.

Keywords
compact locally symmetric space, Poisson transform, spectral correspondence, Weyl chamber flow
Mathematical Subject Classification
Primary: 22E46, 37C85, 37D20, 43A90, 58J50
Secondary: 58J40
Milestones
Received: 16 March 2021
Revised: 7 March 2022
Accepted: 28 May 2022
Published: 11 December 2023
Authors
Joachim Hilgert
Institut für Mathematik
Universität Paderborn
Paderborn
Germany
Tobias Weich
Institut für Mathematik
Universität Paderborn
Paderborn
Germany
Lasse L. Wolf
Institut für Mathematik
Universität Paderborn
Paderborn
Germany

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